Research

38 pages, 2536 KiB

Open AccessEditor’s ChoiceArticle

Projected-Reflected Subgradient-Extragradient Method and Its Real-World Applications

by Aviv Gibali, Olaniyi S. Iyiola, Lanre Akinyemi and Yekini Shehu

Symmetry 2021, 13(3), 489; https://doi.org/10.3390/sym13030489 - 16 Mar 2021

Cited by 8 | Viewed by 2026

Our main focus in this work is the classical variational inequality problem with Lipschitz continuous and pseudo-monotone mapping in real Hilbert spaces. An adaptive reflected subgradient-extragradient method is presented along with its weak convergence analysis. The novelty of the proposed method lies in [...] Read more.

Our main focus in this work is the classical variational inequality problem with Lipschitz continuous and pseudo-monotone mapping in real Hilbert spaces. An adaptive reflected subgradient-extragradient method is presented along with its weak convergence analysis. The novelty of the proposed method lies in the fact that only one projection onto the feasible set in each iteration is required, and there is no need to know/approximate the Lipschitz constant of the cost function a priori. To illustrate and emphasize the potential applicability of the new scheme, several numerical experiments and comparisons in tomography reconstruction, Nash–Cournot oligopolistic equilibrium, and more are presented. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems)

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14 pages, 294 KiB

Open AccessArticle

Monotone Iterative Technique for the Periodic Solutions of High-Order Delayed Differential Equations in Abstract Spaces

by He Yang and Yongxiang Li

Symmetry 2021, 13(3), 449; https://doi.org/10.3390/sym13030449 - 10 Mar 2021

Viewed by 1250

Abstract

This paper deals with the existence of

ω

-periodic solutions for nth-order ordinary differential equation involving fixed delay in Banach space E. [...] Read more.

This paper deals with the existence of

ω

-periodic solutions for nth-order ordinary differential equation involving fixed delay in Banach space E.

L_{n} u (t) = f (t, u (t), u (t - τ)), t \in R,

where

L_{n} u (t) : = u^{(n)} (t) + \sum_{i = 0}^{n - 1} a_{i} u^{(i)} (t)

,

a_{i} \in R

,

i = 0, 1, \dots, n - 1

, are constants,

f (t, x, y) : R \times E \times E ⟶ E

is continuous and

ω

-periodic with respect to t,

τ > 0

. By applying the approach of upper and lower solutions and the monotone iterative technique, some existence and uniqueness theorems are proved under essential conditions. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems)

7 pages, 260 KiB

Open AccessArticle

A Note on the Periodic Solutions for a Class of Third Order Differential Equations

by Zouhair Diab, Juan L. G. Guirao and Juan A. Vera

Symmetry 2021, 13(1), 31; https://doi.org/10.3390/sym13010031 - 27 Dec 2020

Cited by 4 | Viewed by 1543

Abstract

The aim of the present work is to study the necessary and sufficient conditions for the existence of periodic solutions for a class of third order differential equations by using the averaging theory. Moreover, we use the symmetry of the Monodromy matrix to [...] Read more.

The aim of the present work is to study the necessary and sufficient conditions for the existence of periodic solutions for a class of third order differential equations by using the averaging theory. Moreover, we use the symmetry of the Monodromy matrix to study the stability of these solutions. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems)

13 pages, 343 KiB

Open AccessArticle

Global Stabilization of a Reaction Wheel Pendulum: A Discrete-Inverse Optimal Formulation Approach via A Control Lyapunov Function

by Oscar Danilo Montoya, Walter Gil-González, Juan A. Dominguez-Jimenez, Alexander Molina-Cabrera and Diego A. Giral-Ramírez

Symmetry 2020, 12(11), 1771; https://doi.org/10.3390/sym12111771 - 26 Oct 2020

Cited by 2 | Viewed by 3534

Abstract

This paper deals with the global stabilization of the reaction wheel pendulum (RWP) in the discrete-time domain. The discrete-inverse optimal control approach via a control Lyapunov function (CLF) is employed to make the stabilization task. The main advantages of using this control methodology [...] Read more.

This paper deals with the global stabilization of the reaction wheel pendulum (RWP) in the discrete-time domain. The discrete-inverse optimal control approach via a control Lyapunov function (CLF) is employed to make the stabilization task. The main advantages of using this control methodology can be summarized as follows: (i) it guarantees exponential stability in closed-loop operation, and (ii) the inverse control law is optimal since it minimizes the cost functional of the system. Numerical simulations demonstrate that the RWP is stabilized with the discrete-inverse optimal control approach via a CLF with different settling times as a function of the control gains. Furthermore, parametric uncertainties and comparisons with nonlinear controllers such as passivity-based and Lyapunov-based approaches developed in the continuous-time domain have demonstrated the superiority of the proposed discrete control approach. All of these simulations have been implemented in the MATLAB software. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems)

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12 pages, 573 KiB

Open AccessArticle

Nonstandard Finite Difference Schemes for the Study of the Dynamics of the Babesiosis Disease

by Quang A. Dang, Manh T. Hoang, Deccy Y. Trejos and Jose C. Valverde

Symmetry 2020, 12(9), 1447; https://doi.org/10.3390/sym12091447 - 02 Sep 2020

Cited by 1 | Viewed by 1600

Abstract

In this paper, a discrete-time model for Babesiosis disease, given by means of nonstandard finite difference (NSFD) schemes, is first provided and analyzed. Mathematical analyses show that the provided NSFD schemes preserve the essential (qualitative) dynamical properties of the continuous-time model, namely, positivity [...] Read more.

In this paper, a discrete-time model for Babesiosis disease, given by means of nonstandard finite difference (NSFD) schemes, is first provided and analyzed. Mathematical analyses show that the provided NSFD schemes preserve the essential (qualitative) dynamical properties of the continuous-time model, namely, positivity and boundedness of the solutions, equilibria, and their stability properties. In particular, the global stability of the disease free equilibrium point is proved by using an appropriate Lyapunov function. As a relevant consequence, we get the dynamic consistency of NSFD schemes in relation to the continuous-time model. Numerical simulations are presented to support the validity of the established theoretical results. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems)

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21 pages, 1469 KiB

Open AccessArticle

Controlling the Perturbations of Solar Radiation Pressure on the Lorentz Spacecraft

by A. Mostafa, M. I. El-Saftawy, Elbaz I. Abouelmagd and Miguel A. López

Symmetry 2020, 12(9), 1423; https://doi.org/10.3390/sym12091423 - 27 Aug 2020

Cited by 5 | Viewed by 2265

Abstract

The aim of the present paper is to analyze the viability of using Lorentz Force (LF) acting on a charged spacecraft to neutralize the effects of Solar Radiation Pressure (SRP) on the longitude of the ascending node and the argument of perigee of [...] Read more.

The aim of the present paper is to analyze the viability of using Lorentz Force (LF) acting on a charged spacecraft to neutralize the effects of Solar Radiation Pressure (SRP) on the longitude of the ascending node and the argument of perigee of the spacecraft’s orbit. In this setting, the Gauss planetary equations for LF and SRP are presented and averaged over the true anomaly. The averaged variations for the longitude of the ascending node (h) and the argument of perigee (g) are invariant under the symmetry

(i, g) ⟶ (- i, - g)

due to Lorentz Force. The sum of change rates due to both perturbing forces of LF and SRP is assigned by zero to estimate the charge amount to balance the variation for the argument of perigee and longitude of ascending. Numerical investigations have been developed to show the evolution of the charge quantity for different orbital parameters at both Low Earth and Geosynchronous Orbits. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems)

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19 pages, 4647 KiB

Open AccessArticle

Analysis of Chaotic Response of Frenkel-Kontorova-Tomlinson Model

by Joaquín Solano Ramírez, Francisco Balibrea Gallego, José Andrés Moreno Nicolás and Fulgencio Marín García

Symmetry 2020, 12(9), 1413; https://doi.org/10.3390/sym12091413 - 25 Aug 2020

Cited by 4 | Viewed by 2064

Abstract

The Frenkel-Kontorova-Tomlinson (FKT) model represents mechanical systems in which the atomic smooth surfaces of two bodies slide against each other. The model is very sensitive to changes of the system parameters, and ranges from simple stable harmonic to chaotic solutions. The design of [...] Read more.

The Frenkel-Kontorova-Tomlinson (FKT) model represents mechanical systems in which the atomic smooth surfaces of two bodies slide against each other. The model is very sensitive to changes of the system parameters, and ranges from simple stable harmonic to chaotic solutions. The design of the model between two bodies for the dynamic problem, following the network method rules, is explained with precision and run on standard electrical circuit simulation software. It provides the phase diagrams of atom displacement for each atom and the total friction force by the summation of all the atom displacements. This article is focused on studying the effect of the selected time step on the result and in the lack of sensitivity of Lyapunov exponents to assess chaotic behaviour. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems)

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12 pages, 408 KiB

Open AccessFeature PaperArticle

Discrete-Inverse Optimal Control Applied to the Ball and Beam Dynamical System: A Passivity-Based Control Approach

by Oscar Danilo Montoya, Walter Gil-González and Carlos Ramírez-Vanegas

Symmetry 2020, 12(8), 1359; https://doi.org/10.3390/sym12081359 - 14 Aug 2020

Cited by 10 | Viewed by 2792

Abstract

This express brief deals with the problem of the state variables regulation in the ball and beam system by applying the discrete-inverse optimal control approach. The ball and beam system model is defined by a set of four-order nonlinear differential equations that are [...] Read more.

This express brief deals with the problem of the state variables regulation in the ball and beam system by applying the discrete-inverse optimal control approach. The ball and beam system model is defined by a set of four-order nonlinear differential equations that are discretized using the forward difference method. The main advantages of using the discrete-inverse optimal control to regulate state variables in dynamic systems are (i) the control input is an optimal signal as it guarantees the minimum of the Hamiltonian function, (ii) the control signal makes the dynamical system passive, and (iii) the control input ensures asymptotic stability in the sense of Lyapunov. Numerical simulations in the MATLAB environment allow demonstrating the effectiveness and robustness of the studied control design for state variables regulation with a wide gamma of dynamic behaviors as a function of the assigned control gains. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems)

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11 pages, 2851 KiB

Open AccessArticle

Anatomical Considerations and Study of the Fractal Dimension around the Posterior Superior Alveolar Artery

by Yolanda Guerrero-Sánchez, Francisco José Gómez García, Manuel Fernández-Martínez, Blanca Pallarés Martínez and Pia López-Jornet

Symmetry 2020, 12(7), 1177; https://doi.org/10.3390/sym12071177 - 16 Jul 2020

Cited by 1 | Viewed by 1899

Abstract

The Posterior Superior Alveolar Artery (PSAA) provides vascular support to molars, gingiva, and maxillary sinus. A tear of the PSAA may cause profuse hemorrhages which may lead to complications at a surgical level. As such, it becomes crucial to anatomically analyse several features [...] Read more.

The Posterior Superior Alveolar Artery (PSAA) provides vascular support to molars, gingiva, and maxillary sinus. A tear of the PSAA may cause profuse hemorrhages which may lead to complications at a surgical level. As such, it becomes crucial to anatomically analyse several features regarding the PSAA as well as the area surrounding it. In this paper, we are particularly interested in the study of the complexity of the periodontal tissue structure which appears close to the location of the PSAA. A total amount of 400 cone beam computed tomography (CBCT) scans (two per subject) were performed to explore the presence of the PSAA, the thickness of the Schneider’s membrane, and the existence of septa. Several parameters were evaluated including the location of the artery in the maxillary sinus, the distance from the PSAA to the alveolar ridge, the thickness of the membrane, the diameter of the cavities produced by the septa, and the fractal dimension of the trabecular tissue that surrounds the PSAA. They were found strong linear relationships between Distal and Central Measures (a Pearson’s

R^{2} = 0.9952

), Mesial and Central Measures (

R^{2} = 0.9950

), and Distal and Mesial Measure (

R^{2} = 0.997

). We hypothesised that the loss of dental pieces would imply a distinct complexity of the trabecular tissue structure surrounding the PSAA. In this way, a p-value equal to

0.001

was provided by the Mann-Whitney test, which supports our hypothesis. Furthermore, the mean of the fractal dimensions of the group of edentulous patients (equal to

1.56

) was found to be lower than the one of the group of non-edentulous patients (equal to

1.61

) with small standard deviations in both cases. Our study suggests that accurate calculations of the fractal dimension combined with the use of CBCT do provide valuable information regarding the area that surrounds the PSAA. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems)

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19 pages, 15166 KiB

Open AccessArticle

A Multistable Chaotic Jerk System with Coexisting and Hidden Attractors: Dynamical and Complexity Analysis, FPGA-Based Realization, and Chaos Stabilization Using a Robust Controller

by Heng Chen, Shaobo He, Ana Dalia Pano Azucena, Amin Yousefpour, Hadi Jahanshahi, Miguel A. López and Raúl Alcaraz

Symmetry 2020, 12(4), 569; https://doi.org/10.3390/sym12040569 - 05 Apr 2020

Cited by 28 | Viewed by 2710

Abstract

In the present work, a new nonequilibrium four-dimensional chaotic jerk system is presented. The proposed system includes only one constant term and has coexisting and hidden attractors. Firstly, the dynamical behavior of the system is investigated using bifurcation diagrams and Lyapunov exponents. It [...] Read more.

In the present work, a new nonequilibrium four-dimensional chaotic jerk system is presented. The proposed system includes only one constant term and has coexisting and hidden attractors. Firstly, the dynamical behavior of the system is investigated using bifurcation diagrams and Lyapunov exponents. It is illustrated that this system either possesses symmetric equilibrium points or does not possess an equilibrium. Rich dynamics are found by varying system parameters. It is shown that the system enters chaos through experiencing a cascade of period doublings, and the existence of chaos is verified. Then, coexisting and hidden chaotic attractors are observed, and basin attraction is plotted. Moreover, using the multiscale C0 algorithm, the complexity of the system is investigated, and a broad area of high complexity is displayed in the parameter planes. In addition, the chaotic behavior of the system is studied by field-programmable gate array implementation. A novel methodology to discretize, simulate, and implement the proposed system is presented, and the successful implementation of the proposed system on FPGA is verified through the simulation outcome. Finally, a robust sliding mode controller is designed to suppress the chaotic behavior of the system. To deal with unexpected disturbances and uncertainties, a disturbance observer is developed along with the designed controller. To show the successful performance of the designed control scheme, numerical simulations are also presented. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems)

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21 pages, 853 KiB

Open AccessArticle

Hybrid Ćirić Type Graphic Υ,Λ-Contraction Mappings with Applications to Electric Circuit and Fractional Differential Equations

by Eskandar Ameer, Hassen Aydi, Muhammad Arshad and Manuel De la Sen

Symmetry 2020, 12(3), 467; https://doi.org/10.3390/sym12030467 - 16 Mar 2020

Cited by 59 | Viewed by 3572

Abstract

In this paper, we initiate the notion of Ćirić type rational graphic

(Υ, Λ)

-contraction pair mappings and provide some new related common fixed point results on partial b-metric spaces endowed with a directed graph G. We also give examples [...] Read more.

In this paper, we initiate the notion of Ćirić type rational graphic

(Υ, Λ)

-contraction pair mappings and provide some new related common fixed point results on partial b-metric spaces endowed with a directed graph G. We also give examples to illustrate our main results. Moreover, we present some applications on electric circuit equations and fractional differential equations. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems)

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7 pages, 233 KiB

Open AccessArticle

Generalized Symmetries and mCK Method Analysis of the (2+1)-Dimensional Coupled Burgers Equations

by Gangwei Wang, Yixing Liu, Shuxin Han, Hua Wang and Xing Su

Symmetry 2019, 11(12), 1473; https://doi.org/10.3390/sym11121473 - 03 Dec 2019

Cited by 8 | Viewed by 1562

Abstract

In this paper, generalized symmetries and mCK method are employed to analyze the (2+1)-dimensional coupled Burgers equations. Firstly, based on the generalized symmetries method, the corresponding symmetries of the (2+1)-dimensional coupled Burgers equations are derived. And then, using the mCK method, symmetry transformation [...] Read more.

In this paper, generalized symmetries and mCK method are employed to analyze the (2+1)-dimensional coupled Burgers equations. Firstly, based on the generalized symmetries method, the corresponding symmetries of the (2+1)-dimensional coupled Burgers equations are derived. And then, using the mCK method, symmetry transformation group theorem is presented. From symmetry transformation group theorem, a great many of new solutions can be derived. Lastly, Lie algebra for given symmetry group are considered. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems)

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